Glossary

8.4 Combining forecasts

2 min read•july 22, 2024

Combining forecasts is a smart way to make predictions more accurate and reliable. By using multiple models, we can capture different aspects of the data and reduce the risk of relying on just one approach.

There are various methods for combining forecasts, from simple averaging to weighted techniques based on model performance. These approaches help create more robust predictions that are less sensitive to outliers or unusual patterns in the data.

Combining Forecasts

Rationale for forecast combination

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  • Improves forecast accuracy by capturing different aspects of time series data with each model
  • Reduces risk of relying on a single model that may be misspecified or perform poorly
  • Provides a more robust and stable forecast less sensitive to outliers or unusual patterns (sudden spikes or dips)

Simple averaging of forecasts

  • Takes arithmetic mean of forecasts from different models
    • Combined Forecast=1n∑i=1nForecasti\text{Combined Forecast} = \frac{1}{n} \sum_{i=1}^{n} \text{Forecast}_i, where nn is the number of models (ARIMA, exponential smoothing)
  • Assigns equal weights to all models assuming they contribute equally to combined forecast
  • Easy to implement and interpret without requiring additional information about model performance or complexity

Weighted averaging techniques

  • Assigns different weights to each model based on historical performance with better performing models receiving higher weights
  • Inverse Mean Squared Error (MSE) weighting:
    1. Calculate MSEi\text{MSE}_i for each model ii
    2. Compute weights as Weighti=1/MSEi∑j=1n1/MSEj\text{Weight}_i = \frac{1/\text{MSE}_i}{\sum_{j=1}^{n} 1/\text{MSE}_j}
    3. Models with lower MSE receive higher weights
  • Akaike Information Criterion (AIC) weighting:
    1. Calculate AICi\text{AIC}_i for each model ii
    2. Compute weights as Weighti=exp⁡(−0.5×AICi)∑j=1nexp⁡(−0.5×AICj)\text{Weight}_i = \frac{\exp(-0.5 \times \text{AIC}_i)}{\sum_{j=1}^{n} \exp(-0.5 \times \text{AIC}_j)}
    3. Models with lower AIC receive higher weights
  • Incorporates model performance information into combination process (accuracy, parsimony)

Accuracy of combined vs individual forecasts

  • Calculate forecast accuracy metrics for combined and individual model forecasts
    • (MAE), Mean Squared Error (MSE), Mean Absolute Percentage Error (MAPE)
  • Compare accuracy metrics of combined forecast with individual models to determine if combined outperforms
  • Assess stability and robustness of combined forecast across different time periods or data subsets (training, validation, testing)
  • Consider trade-off between forecast accuracy and computational complexity as combining may require additional resources and time

Key Terms to Review (7)

Bontemps: Bontemps refers to a statistical concept related to the assessment and combination of different forecasts, typically focusing on achieving optimal predictive accuracy. It emphasizes how multiple forecast models can be integrated to produce a more reliable overall prediction, thereby reducing the impact of individual model biases or errors. The essence of bontemps is to recognize that by combining forecasts, one can harness the strengths of various approaches to enhance decision-making and forecasting precision.
Economic forecasting: Economic forecasting is the process of predicting future economic conditions and trends based on historical data and statistical methods. This practice is crucial for informing decisions in various sectors, such as business planning, policy-making, and investment strategies, by providing insights into potential future developments. Accurate forecasts are essential as they can significantly influence economic performance, helping to mitigate risks and seize opportunities.
Ensemble forecasting: Ensemble forecasting is a technique used in statistical modeling and time series analysis that involves generating multiple forecasts from different models or simulations to capture the uncertainty in predictions. By combining the results from these various forecasts, ensemble forecasting aims to improve overall prediction accuracy and provide a more comprehensive view of potential outcomes, making it a valuable tool in decision-making processes.
Mean Absolute Error: Mean Absolute Error (MAE) is a measure of the average magnitude of errors in a set of forecasts, without considering their direction. It quantifies how far predictions deviate from actual values by averaging the absolute differences between predicted and observed values. This concept is essential for evaluating the accuracy of various forecasting methods and models, as it provides a straightforward metric for comparing performance across different time series analysis techniques.
Pooling forecasts: Pooling forecasts refers to the practice of combining multiple forecasting methods or models to create a single, more accurate prediction. This technique leverages the strengths of different approaches, reducing the overall forecast error and improving reliability. By integrating various forecasts, pooling can capture different aspects of the data and provide a more robust outcome than any individual forecast method.
Root Mean Square Error: Root Mean Square Error (RMSE) is a widely used metric that measures the average magnitude of the errors between predicted and observed values, calculated as the square root of the average of squared differences. This metric provides a clear representation of how well a forecasting model is performing, allowing for comparisons across different methods and scenarios. RMSE is essential in evaluating forecast accuracy, particularly when combining forecasts, creating point forecasts and prediction intervals, or applying trend methods and decomposition techniques.
Weighted average: A weighted average is a mean that takes into account the relative importance or frequency of different values in a dataset, giving more weight to certain numbers over others. This method helps to provide a more accurate representation of the data, especially when some values are more significant than others in a forecasting context. The concept is essential in combining different forecasts or applying simple exponential smoothing to time series data.
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